Theorem alrim3con13v 38743

Description: Closed form of alrimi 2082 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 39087 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
alrim3con13v	|- ( ( ph -> A. x ph ) -> ( ( ps /\ ph /\ ch ) -> A. x ( ps /\ ph /\ ch ) ) )

Distinct variable groups:   ps, x   ch, x

Allowed substitution hint:   ph( x)

Proof of Theorem alrim3con13v

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . 5 |- ( ( ps /\ ph /\ ch ) -> ps )
2	1	a1i 11	. . . 4 |- ( ( ph -> A. x ph ) -> ( ( ps /\ ph /\ ch ) -> ps ) )
3		ax-5 1839	. . . 4 |- ( ps -> A. x ps )
4	2, 3	syl6 35	. . 3 |- ( ( ph -> A. x ph ) -> ( ( ps /\ ph /\ ch ) -> A. x ps ) )
5		simp2 1062	. . . 4 |- ( ( ps /\ ph /\ ch ) -> ph )
6	5	imim1i 63	. . 3 |- ( ( ph -> A. x ph ) -> ( ( ps /\ ph /\ ch ) -> A. x ph ) )
7		simp3 1063	. . . . 5 |- ( ( ps /\ ph /\ ch ) -> ch )
8	7	a1i 11	. . . 4 |- ( ( ph -> A. x ph ) -> ( ( ps /\ ph /\ ch ) -> ch ) )
9		ax-5 1839	. . . 4 |- ( ch -> A. x ch )
10	8, 9	syl6 35	. . 3 |- ( ( ph -> A. x ph ) -> ( ( ps /\ ph /\ ch ) -> A. x ch ) )
11	4, 6, 10	3jcad 1243	. 2 |- ( ( ph -> A. x ph ) -> ( ( ps /\ ph /\ ch ) -> ( A. x ps /\ A. x ph /\ A. x ch ) ) )
12		19.26-3an 1800	. 2 |- ( A. x ( ps /\ ph /\ ch ) <-> ( A. x ps /\ A. x ph /\ A. x ch ) )
13	11, 12	syl6ibr 242	1 |- ( ( ph -> A. x ph ) -> ( ( ps /\ ph /\ ch ) -> A. x ( ps /\ ph /\ ch ) ) )

Syntax hints:   -> wi 4   /\ w3a 1037   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039

This theorem is referenced by:  tratrbVD  39097